Then a linear map is called a symplectic map if the pullback preserves the symplectic form, i. e., where the pullback form is defined by.

A local trivialization of a principal bundle P is given by a section s of P over an open subset U of M. Then the pullback s < sup >*</ sup > ω of a principal connection is a 1-form on U with values in.

Let φ: M → N be a smooth map between ( smooth ) manifolds M and N, and suppose f: N → R is a smooth function on N. Then the pullback of f by φ is the smooth function φ < sup >*</ sup > f on M defined by

Then the pullback Φ < sup >*</ sup > F of F by Φ is a multilinear form on V defined by precomposing F with Φ.

Then the differential of φ, φ < sub >*</ sub > = dφ ( or Dφ ), is a vector bundle morphism ( over M ) from the tangent bundle TM of M to the pullback bundle φ < sup >*</ sup > TN.

Then a pullback of f and g ( in Set ) is given by the preimage f < sup >- 1 </ sup > together with the inclusion of the preimage in A

Denote this Thom class by U. Then for a class b in the cohomology of the base, we can compute the Thom isomorphism via the pullback of the bundle projection and the cohomology cup product:

Then the joint distribution of X and Y is completely determined by our channel and by our choice of, the marginal distribution of messages we choose to send over the channel.

Then, if guilt is determined, they decide the appropriate penalty.

Then, after putting the matter to God in prayer, he determined that the first person to enter his room the next morning should be the next Emperor, and that person was Justin, the chief of his guards.

Then, in 1846, the element ilmenium was claimed to have been discovered, but later was determined to be impure niobium.

Then we imagine the rays are parallel ( and given by two angles ), and we let the colour of a point on the surface be determined by the angle between this direction and the slope of the surface at the point.

Then, by projecting the muzzle on the screen with the measured angles the impact point is determined.

Then Y is completely determined by X, so that X and Y are perfectly dependent, but their correlation is zero ; they are uncorrelated.

Then the set of rational points on C may be determined as follows:

Then, a tail event is an event whose occurrence or failure is determined by the values of these random variables but which is probabilistically independent of each finite subset of these random variables.

Then, using the formula, the asymptotic behavior can be determined as follows:

Then, if the assembly was composed of Tribes, the order of the vote had to be determined.

Then Yājñavalkya determined not to have any human guru thereafter.

Then, in 1964, thanks to Stanley Dance, Earl Hines ' determined friend and unofficial manager, Hines was " suddenly rediscovered " following a series of " recitals " at The Little Theatre in New York that Dance had cajoled him into.

The frame bundle F ( E ) can be given a natural topology and bundle structure determined by that of E. Let ( U < sub > i </ sub >, φ < sub > i </ sub >) be a local trivialization of E. Then for each x ∈ U < sub > i </ sub > one has a linear isomorphism φ < sub > i, x </ sub >: E < sub > x </ sub > → R < sup > k </ sup >.

Then a reducible representation of the bonding is determined demonstrated below for water:

Then determined who should be party ’ s declaratory.

Then in 402, he decided to launch an attack on Later Yan itself, determined to protect his Kingdom from further threat.

Then the normal stress and shear stress as a function of the principal stresses can be determined by making.

Then the closed subscheme determined by the base change is the closed image of X ′.< ref > EGA IV < sub > 2 </ sub >, Proposition 2. 3. 2 .</ ref >

Then the flow of net investment per unit of time is determined by balancing losses from having a less-than-optimal level of capital with the adjustment costs of installing new capital ; these adjustment costs in per-unit terms may be an increasing function of the speed of installation.

Then armies and military formations grew, so needed more lodgement area to organize a force large enough to stage a break out against a determined enemy, and again the technical meaning of the term expanded, again referring to a large fortified area about at bridge end.

Then a final vote is taken and the number of delegates for each candidate is determined, proportional to his or her share of the vote.

Then in 2000, Dayton-Hudson renamed itself Target Corporation, having determined that Target was where more of its future growth would be and the more nationally visible division to stockholders.

Then, expecting to be overwhelmed by British reinforcements, they quickly turned back to Fort Erie where they fought a second battle, the Battle of Fort Erie ( 1866 ), against a small but determined detachment of Canadians holding the town.

Then she turned back to Wilson and smiled, and he wasn't quite sure what she meant by it.

Then he astonished Matsuo by pushing and dragging himself until he sat.

Then, after overtures to accept a settlement and go through with a divorce, Miriam gave a ghastly echo of Mrs. Micawber by suddenly stating, `` I will never leave Mr. Wright ''.

Then Miriam varied the senseless psychological warfare by suddenly withdrawing a suit for separate maintenance that had been pending, and asking for divorce on the grounds of cruelty, with the understanding that Wright would not contest it.

My own stern hand has rent the ancient bond, And thereof shall the ending not have end: But not for me, that loved her, to be fond Lightly to please me with a newer friend Then hold it more than bravest-feathered song, That I affirm to thee, with heart of pride, I knew not what did to a friend belong Till I stood up, true friend, by thy true side ; ;

Certainly, the meaning is clearer to one who is not familiar with Biblical teachings, in the New English Bible which reads: `` Then Jesus arrived at Jordan from Galilee, and he came to John to be baptized by him.

Then came their bathroom, and then a bedroom that, judging by the photographs on the walls, must belong to Mme Cestre.

Then epistolatory me was a foreign correspondent dispatching exciting cables and communiques, full of dash and wit and glamor, quoting from the books I read, imitating the grand styles of the authors recommended by a teacher in whose special, after-school class I was enrolled.

Then, at last, his day would fall into an ordered pattern and he would be free to read, or garden or just wander through the woods in the late afternoon, accompanied by his dogs.

Then there are the trustees and officers of the great educational foundations, who inevitably exert an influence on educational decisions by their support or refusal to support various educational programs, experiments, and demonstrations.

Then he might finally recognize it, apparently by combining the visual blot, actually being seen, with tactual feelings in the head or body accompanying the tracing movements.

Then, advised by the Architect of the Capitol, the Joint Committee for the Library, traditionally responsible for the works of art in the building, ordered the space cleared and painted in fresco, to show `` the Peace after the Civil War '', `` the Spanish-American War '', and `` the Birth of Aviation '', to match as nearly as feasible Brumidi's technique and composition.

Then, by squeezing her foot three times, he came up -- presto -- with a different diagnosis with each squeeze.

Then he asked to use the phone and called Brian Thayer, who said that he was just leaving to keep a lunch date but would be home by two o'clock.

Then, when the case went to the jury, the judge excused one of the jurors, saying the juror had told him he had been accosted by masked men at his motel the night before the trial opened.

Then Charlie spooned out some quick impressions of the Nikita he had glimpsed: `` I was captivated by his humor, frankness and good nature and by his kind, strong and somewhat sly face ''.

Then, if the middle number is activated to its greatest potential in terms of this square, through multiplying it by the highest number, 9 ( which is the square of the base number ), the result is 45 ; ;

Then there are a pair of old biddies played by Grace Carney and Sibly Bowan who may be right off the shelf of stock Irish characters, but they put such a combination of good will and malevolence into their parts that they're quite entertaining.

Then, in order to prevent Polk from dissipating his forces by implementing his proposal to allow some men to join a partisan group, Johnston ordered him to send Pillow and 5, 000 men to Fort Donelson.

Seven stories are inspired by a nursery rhyme: And Then There Were None by Ten Little Indians ; One, Two, Buckle My Shoe by One, Two, Buckle My Shoe ; Five Little Pigs by This Little Piggy ; Crooked House by There Was a Crooked Man ; A Pocket Full of Rye by Sing a Song of Sixpence ; Hickory Dickory Dock by Hickory Dickory Dock, and Three Blind Mice by Three Blind Mice.